# Properties

 Label 10.9.c.b Level 10 Weight 9 Character orbit 10.c Analytic conductor 4.074 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$10 = 2 \cdot 5$$ Weight: $$k$$ = $$9$$ Character orbit: $$[\chi]$$ = 10.c (of order $$4$$ and degree $$2$$)

## Newform invariants

 Self dual: No Analytic conductor: $$4.07378610061$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{601})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 8 + 8 \beta_{1} ) q^{2} + ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3} + 128 \beta_{1} q^{4} + ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} + ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7} + ( -1024 + 1024 \beta_{1} ) q^{8} + ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( 8 + 8 \beta_{1} ) q^{2} + ( 22 - 21 \beta_{1} + \beta_{3} ) q^{3} + 128 \beta_{1} q^{4} + ( -216 - 71 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} ) q^{5} + ( 352 + 8 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{6} + ( 1442 + 1442 \beta_{1} - 21 \beta_{2} ) q^{7} + ( -1024 + 1024 \beta_{1} ) q^{8} + ( -1876 \beta_{1} + 43 \beta_{2} + 43 \beta_{3} ) q^{9} + ( -1112 - 2272 \beta_{1} - 24 \beta_{2} + 72 \beta_{3} ) q^{10} + ( -3824 - 141 \beta_{1} + 141 \beta_{2} - 141 \beta_{3} ) q^{11} + ( 2816 + 2816 \beta_{1} - 128 \beta_{2} ) q^{12} + ( 11235 - 11553 \beta_{1} - 318 \beta_{3} ) q^{13} + ( 22904 \beta_{1} - 168 \beta_{2} - 168 \beta_{3} ) q^{14} + ( -28982 - 42167 \beta_{1} + 266 \beta_{2} - 153 \beta_{3} ) q^{15} -16384 q^{16} + ( 871 + 871 \beta_{1} + 66 \beta_{2} ) q^{17} + ( 15352 - 14664 \beta_{1} + 688 \beta_{3} ) q^{18} + ( 122890 \beta_{1} + 438 \beta_{2} + 438 \beta_{3} ) q^{19} + ( 9856 - 27264 \beta_{1} - 768 \beta_{2} + 384 \beta_{3} ) q^{20} + ( 221200 + 1883 \beta_{1} - 1883 \beta_{2} + 1883 \beta_{3} ) q^{21} + ( -30592 - 30592 \beta_{1} + 2256 \beta_{2} ) q^{22} + ( -113258 + 115139 \beta_{1} + 1881 \beta_{3} ) q^{23} + ( 44032 \beta_{1} - 1024 \beta_{2} - 1024 \beta_{3} ) q^{24} + ( -229705 - 172605 \beta_{1} - 435 \beta_{2} - 3045 \beta_{3} ) q^{25} + ( 179760 - 2544 \beta_{1} + 2544 \beta_{2} - 2544 \beta_{3} ) q^{26} + ( -220892 - 220892 \beta_{1} - 2836 \beta_{2} ) q^{27} + ( -184576 + 181888 \beta_{1} - 2688 \beta_{3} ) q^{28} + ( 132400 \beta_{1} + 96 \beta_{2} + 96 \beta_{3} ) q^{29} + ( 104256 - 567064 \beta_{1} + 3352 \beta_{2} + 904 \beta_{3} ) q^{30} + ( 1165300 - 2787 \beta_{1} + 2787 \beta_{2} - 2787 \beta_{3} ) q^{31} + ( -131072 - 131072 \beta_{1} ) q^{32} + ( -1143320 + 1133574 \beta_{1} - 9746 \beta_{3} ) q^{33} + ( 14464 \beta_{1} + 528 \beta_{2} + 528 \beta_{3} ) q^{34} + ( 746074 - 881356 \beta_{1} + 273 \beta_{2} + 14406 \beta_{3} ) q^{35} + ( 245632 + 5504 \beta_{1} - 5504 \beta_{2} + 5504 \beta_{3} ) q^{36} + ( -1386939 - 1386939 \beta_{1} - 3564 \beta_{2} ) q^{37} + ( -979616 + 986624 \beta_{1} + 7008 \beta_{3} ) q^{38} + ( 1899033 \beta_{1} + 4557 \beta_{2} + 4557 \beta_{3} ) q^{39} + ( 300032 - 145408 \beta_{1} - 9216 \beta_{2} - 3072 \beta_{3} ) q^{40} + ( 2677024 - 7653 \beta_{1} + 7653 \beta_{2} - 7653 \beta_{3} ) q^{41} + ( 1769600 + 1769600 \beta_{1} - 30128 \beta_{2} ) q^{42} + ( 1243038 - 1213605 \beta_{1} + 29433 \beta_{3} ) q^{43} + ( -471424 \beta_{1} + 18048 \beta_{2} + 18048 \beta_{3} ) q^{44} + ( -3054907 - 572642 \beta_{1} + 5021 \beta_{2} - 18098 \beta_{3} ) q^{45} + ( -1812128 + 15048 \beta_{1} - 15048 \beta_{2} + 15048 \beta_{3} ) q^{46} + ( -1398618 - 1398618 \beta_{1} + 72999 \beta_{2} ) q^{47} + ( -360448 + 344064 \beta_{1} - 16384 \beta_{3} ) q^{48} + ( 1646596 \beta_{1} - 60123 \beta_{2} - 60123 \beta_{3} ) q^{49} + ( -481160 - 3221960 \beta_{1} + 20880 \beta_{2} - 27840 \beta_{3} ) q^{50} + ( -457468 - 515 \beta_{1} + 515 \beta_{2} - 515 \beta_{3} ) q^{51} + ( 1438080 + 1438080 \beta_{1} + 40704 \beta_{2} ) q^{52} + ( 5047325 - 5097833 \beta_{1} - 50508 \beta_{3} ) q^{53} + ( -3556960 \beta_{1} - 22688 \beta_{2} - 22688 \beta_{3} ) q^{54} + ( -2351592 + 9825523 \beta_{1} - 51939 \beta_{2} - 1653 \beta_{3} ) q^{55} + ( -2953216 - 21504 \beta_{1} + 21504 \beta_{2} - 21504 \beta_{3} ) q^{56} + ( -596312 - 596312 \beta_{1} - 104056 \beta_{2} ) q^{57} + ( -1058432 + 1059968 \beta_{1} + 1536 \beta_{3} ) q^{58} + ( -172550 \beta_{1} + 127782 \beta_{2} + 127782 \beta_{3} ) q^{59} + ( 5377792 - 3675648 \beta_{1} + 19584 \beta_{2} + 34048 \beta_{3} ) q^{60} + ( -10824672 + 68331 \beta_{1} - 68331 \beta_{2} + 68331 \beta_{3} ) q^{61} + ( 9322400 + 9322400 \beta_{1} + 44592 \beta_{2} ) q^{62} + ( 9550534 - 9388029 \beta_{1} + 162505 \beta_{3} ) q^{63} -2097152 \beta_{1} q^{64} + ( 3874593 + 15997908 \beta_{1} + 79491 \beta_{2} + 103347 \beta_{3} ) q^{65} + ( -18293120 - 77968 \beta_{1} + 77968 \beta_{2} - 77968 \beta_{3} ) q^{66} + ( 2480678 + 2480678 \beta_{1} - 202113 \beta_{2} ) q^{67} + ( -111488 + 119936 \beta_{1} + 8448 \beta_{3} ) q^{68} + ( -9220477 \beta_{1} - 73757 \beta_{2} - 73757 \beta_{3} ) q^{69} + ( 13134688 - 1080072 \beta_{1} - 113064 \beta_{2} + 117432 \beta_{3} ) q^{70} + ( 4993028 - 108699 \beta_{1} + 108699 \beta_{2} - 108699 \beta_{3} ) q^{71} + ( 1965056 + 1965056 \beta_{1} - 88064 \beta_{2} ) q^{72} + ( -3126215 + 2677967 \beta_{1} - 448248 \beta_{3} ) q^{73} + ( -22219536 \beta_{1} - 28512 \beta_{2} - 28512 \beta_{3} ) q^{74} + ( -5516110 + 23912715 \beta_{1} + 96480 \beta_{2} - 284515 \beta_{3} ) q^{75} + ( -15673856 + 56064 \beta_{1} - 56064 \beta_{2} + 56064 \beta_{3} ) q^{76} + ( -27757240 - 27757240 \beta_{1} + 481026 \beta_{2} ) q^{77} + ( -15155808 + 15228720 \beta_{1} + 72912 \beta_{3} ) q^{78} + ( -34376880 \beta_{1} + 102768 \beta_{2} + 102768 \beta_{3} ) q^{79} + ( 3538944 + 1163264 \beta_{1} - 49152 \beta_{2} - 98304 \beta_{3} ) q^{80} + ( 24175343 + 120787 \beta_{1} - 120787 \beta_{2} + 120787 \beta_{3} ) q^{81} + ( 21416192 + 21416192 \beta_{1} + 122448 \beta_{2} ) q^{82} + ( 16164594 - 15967749 \beta_{1} + 196845 \beta_{3} ) q^{83} + ( 28072576 \beta_{1} - 241024 \beta_{2} - 241024 \beta_{3} ) q^{84} + ( -3095821 + 1235524 \beta_{1} - 17067 \beta_{2} + 3351 \beta_{3} ) q^{85} + ( 19888608 + 235464 \beta_{1} - 235464 \beta_{2} + 235464 \beta_{3} ) q^{86} + ( 2189536 + 2189536 \beta_{1} - 128272 \beta_{2} ) q^{87} + ( 3915776 - 3627008 \beta_{1} + 288768 \beta_{3} ) q^{88} + ( 70322120 \beta_{1} - 72360 \beta_{2} - 72360 \beta_{3} ) q^{89} + ( -20002904 - 28980224 \beta_{1} + 184952 \beta_{2} - 104616 \beta_{3} ) q^{90} + ( -17763396 - 215943 \beta_{1} + 215943 \beta_{2} - 215943 \beta_{3} ) q^{91} + ( -14497024 - 14497024 \beta_{1} - 240768 \beta_{2} ) q^{92} + ( 4700656 - 3652410 \beta_{1} + 1048246 \beta_{3} ) q^{93} + ( -21793896 \beta_{1} + 583992 \beta_{2} + 583992 \beta_{3} ) q^{94} + ( -20183500 - 36078750 \beta_{1} - 800850 \beta_{2} + 241650 \beta_{3} ) q^{95} + ( -5767168 - 131072 \beta_{1} + 131072 \beta_{2} - 131072 \beta_{3} ) q^{96} + ( 8470809 + 8470809 \beta_{1} + 115080 \beta_{2} ) q^{97} + ( -13653752 + 12691784 \beta_{1} - 961968 \beta_{3} ) q^{98} + ( 98005883 \beta_{1} - 422885 \beta_{2} - 422885 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{2} + 86q^{3} - 870q^{5} + 1376q^{6} + 5726q^{7} - 4096q^{8} + O(q^{10})$$ $$4q + 32q^{2} + 86q^{3} - 870q^{5} + 1376q^{6} + 5726q^{7} - 4096q^{8} - 4640q^{10} - 14732q^{11} + 11008q^{12} + 45576q^{13} - 115090q^{15} - 65536q^{16} + 3616q^{17} + 60032q^{18} + 37120q^{20} + 877268q^{21} - 117856q^{22} - 456794q^{23} - 913600q^{25} + 729216q^{26} - 889240q^{27} - 732928q^{28} + 421920q^{30} + 4672348q^{31} - 524288q^{32} - 4553788q^{33} + 2956030q^{35} + 960512q^{36} - 5554884q^{37} - 3932480q^{38} + 1187840q^{40} + 10738708q^{41} + 7018144q^{42} + 4913286q^{43} - 12173390q^{45} - 7308704q^{46} - 5448474q^{47} - 1409024q^{48} - 1827200q^{50} - 1827812q^{51} + 5833728q^{52} + 20290316q^{53} - 9506940q^{55} - 11726848q^{56} - 2593360q^{57} - 4236800q^{58} + 21482240q^{60} - 43572012q^{61} + 37378784q^{62} + 37877126q^{63} + 15450660q^{65} - 72860608q^{66} + 9518486q^{67} - 462848q^{68} + 52077760q^{70} + 20406908q^{71} + 7684096q^{72} - 11608364q^{73} - 21302450q^{75} - 62919680q^{76} - 110066908q^{77} - 60769056q^{78} + 14254080q^{80} + 96218224q^{81} + 85909664q^{82} + 64264686q^{83} - 12424120q^{85} + 78612576q^{86} + 8501600q^{87} + 15085568q^{88} - 79432480q^{90} - 70189812q^{91} - 58469632q^{92} + 16706132q^{93} - 82819000q^{95} - 22544384q^{96} + 34113396q^{97} - 52691072q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 301 x^{2} + 22500$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 151 \nu$$$$)/150$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 250 \nu^{2} + 401 \nu + 37650$$$$)/50$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 375 \nu^{2} + 526 \nu - 56475$$$$)/75$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - 5 \beta_{1}$$$$)/10$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{3} + \beta_{2} - \beta_{1} - 1506$$$$)/10$$ $$\nu^{3}$$ $$=$$ $$($$$$-151 \beta_{3} - 151 \beta_{2} + 2255 \beta_{1}$$$$)/10$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$\beta_{1}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 11.7577i 12.7577i 11.7577i − 12.7577i
8.00000 8.00000i −39.7883 39.7883i 128.000i −401.365 479.094i −636.612 144.447 144.447i −1024.00 1024.00i 3394.79i −7043.67 621.836i
3.2 8.00000 8.00000i 82.7883 + 82.7883i 128.000i −33.6352 + 624.094i 1324.61 2718.55 2718.55i −1024.00 1024.00i 7146.79i 4723.67 + 5261.84i
7.1 8.00000 + 8.00000i −39.7883 + 39.7883i 128.000i −401.365 + 479.094i −636.612 144.447 + 144.447i −1024.00 + 1024.00i 3394.79i −7043.67 + 621.836i
7.2 8.00000 + 8.00000i 82.7883 82.7883i 128.000i −33.6352 624.094i 1324.61 2718.55 + 2718.55i −1024.00 + 1024.00i 7146.79i 4723.67 5261.84i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.c Odd 1 yes

## Hecke kernels

This newform can be constructed as the kernel of the linear operator $$T_{3}^{4} - 86 T_{3}^{3} + 3698 T_{3}^{2} + 566568 T_{3} + 43401744$$ acting on $$S_{9}^{\mathrm{new}}(10, [\chi])$$.